[Liste-CICMA] SÉMINAIRE QUÉBEC-VERMONT NUMBER THEORY (28/09/2017, Francesco Baldassari, Djorje Milicevic)

Guillermo Martinez-Zalce martinez at crm.umontreal.ca
Mon Sep 25 10:21:25 EDT 2017




Le jeudi 28 septembre 2017 / Thursday, September 28, 2017

10 h 30 - 12 h / 10:30 a.m. - 12:00 p.m.

Francesco Baldassari (Padova University)

Harmonic functions attached to meromorphic connections on non-archimedean curves.

McGill University, Burnside Hall salle BH920

Let k be a non-archimedean algebraically closed 
eld of characteristic 0. Consider a
compact separated quasi-smooth strictly k-analytic curve X in the sense of Berkovich
and a 
nite set Z  X of k-rational points. Let X be a strictly semistable model of
X such that the points of Z lie in distinct residue classes of smooth components of X.
We consider a connection (E;r) on such a curve, with meromorphic singularities at
points of Z, where E is a coherent and locally free OX-module. Then, at each point of
U = X􀀀Z, we may 
lter the space of local solutions by the (X;Z)-normalized radius of
convergence. This determines a 
nite graph 􀀀X;Z (with edges of possibly in
nite length)
whose complement in X is a disjoint union of open disks of \(X;Z)-normalized radius
1" and leads to the de
nition of a Newton polygon NX;Z (􀀀; (E;r)) called the ((X;Z)-
normalized) convergence polygon of (E;r). We prove, under the technical assumption
that the OX-module E is the generic 
ber of a coherent and locally free OX-module
E, that the convergence polygon, as a function on the curve, is continuous on U. The
so-called spectral part of this polygon is a harmonic function but at the boundary
of U ( i.e. at Z and at the Shilov boundary of X). More precisely, we show that the
convergence (resp. spectral) polygon function factors as the composition of a retraction
onto a 
nite re
nement 􀀀 of the graph 􀀀X;Z followed by a polygon-valued function on
􀀀 which is continuous, piecewise ane with integral slopes (resp. harmonic but at the
vertices of 􀀀X;Z which are boundary points of U). Finally we show that, in a suciently
small neighborhood of any z 2 Z, the Turrittin polygon of the singularity of (E;r) at
z prevails on the convergence polygon.
From joint

Le jeudi 28 septembre 2017 / Thursday, September 28, 2017

14 h - 15 h 30 / 2:00 p.m. - 3:30 p.m.

Djordje Milicevic (Bryn Mawr)

p-adic analytic twists, exponential sums, and strong subconvexity

Concordia University, Library Building, 9th floor, room LB 921-4

Many of the principal analytic questions about L-functions, such as the subconvexity estimates, moment evaluations, and nonvanishing of their critical values, at their core rely on estimates of associated exponential sums. In this talk, we will present several recent estimates for short exponential sums with phases involving p-adically analytic fluctuations. As applications, we obtain subconvexity bounds for Dirichlet and twisted modular L-functions with characters to a high prime power modulus, which are as strong as those available in the t-aspect. From an adelic viewpoint, the analogy between this so-called "depth aspect" and the familiar t-aspect is particularly natural, as one is focusing on ramification at one (finite or infinite) place at a time. Among the tools, we develop p-adic counterparts to Farey dissection, the circle method, and van der Corput estimates. Some of the results are joint work with Valentin Blomer.

Responsable(s) :
Henri Darmon (darmon at math.mcgill.ca)
Adrian Iovita (adrian.iovita at concordia.ca)
Maksym Radziwill (maksym.radziwill at gmail.com)

http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html <http://www.dms.umontreal.ca/~qvnts/QVNTSinfo.html>

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